Rates of information aggregation in common value auctions

We study the rates at which transaction prices aggregate information in common value auctions under the different information structures in Wilson (Rev. Econ. Stud. 44 (1977) 511) and Pesendorfer and Swinkels (Econometrica 65 (1997) 1247). We consider uniform-price auctions in which k identical objects of unknown value are auctioned to n bidders, where both n and k are allowed to diverge to infinity, and k/n converges to a number in [0,1). The Wilson assumptions lead to information aggregation at a rate proportional to , but the price aggregates information at a rate proportional to in the PS setting. We also consider English auctions, and investigate whether the extra information revealed in equilibrium improves convergence rates in these auctions.     

Asymmetric first price auctions

A new approach to asymmetric first price auctions is proposed which circumvents the need to examine bidding strategies directly. Specifically, the ratio of bidders' (endogenous) payoffs is analyzed and compared to the ratio of the (exogenous) distribution functions that describe beliefs. Most of the results are inferred from this comparison. In the existing theoretical literature, assumptions of first order stochastic dominance or stronger imply that the latter ratio has very specific properties, but no such assumptions are imposed here. It is proven that first order stochastic dominance is necessary for bidding strategies not to cross. When this assumption is relaxed in the numerical literature it is done in a manner that leads to exactly one crossing. However, it is straightforward to construct examples with several crossings. Finally, bid distributions will cross in auctions with two bidders whenever second order (but not first order) stochastic dominance applies.     

On ascending Vickrey auctions for heterogeneous objects

We construct an ascending auction for heterogeneous objects by applying a primal-dual algorithm to a linear program that represents the efficient-allocation problem for this setting. The auction assigns personalized prices to bundles, and asks bidders to report their preferred bundles in each round. A bidder's prices are increased when he belongs to a “minimally undersupplied” set of bidders. This concept generalizes the notion of “overdemanded” sets of objects introduced by Demange, Gale, and Sotomayor for the one-to-one assignment problem. Under a submodularity condition, the auction implements the Vickrey-Clarke-Groves outcome; we show that this type of condition is somewhat necessary to do so. When classifying the ascending-auction literature in terms of their underlying algorithms, our auction fills a gap in that literature. We relate our results to various ascending auctions in the literature.     

Asymmetric first-price menu auctions under intricate uncertainty

This paper studies asymmetric first-price menu auctions in the procurement environment where the buyer does not commit to a decision rule and asymmetric sellers have interdependent costs and statistically affiliated signals. Sellers compete in bidding a menu of contracts, where a contract specifies a vector of characteristics and a payment required from the buyer for delivering these characteristics. The buyer does not commit ex-ante to a decision rule but rather upon observing all the menus offered by sellers chooses the best contract. This paper establishes the existence of a continuum of separating monotone equilibria in this game bounded above by the jointly ex-post efficient outcome and below by the jointly interim efficient outcome. It shows that the jointly ex-post efficient equilibrium outcome is the only ex-post renegotiation-proof outcome and it is also ex-ante robust to all continuation equilibria.     

Best-of-three all-pay auctions

We study a best-of-three all-pay auction. It is shown that with values of winning and without values of losing, this auction is less productive (the players’ total expected effort is smaller) than the one-stage all-pay auction. However, with different values of losing over the contest’s stages it may be more productive than the one-stage all-pay auction.     

Asymmetric English auctions

This paper studies equilibria of the n-bidder single-object English, or open ascending price, auction in a setting with interdependent values and asymmetric bidders. Maskin (in: H. Siebert (Ed.), Privatization, Institut fur Weltwirtschaften der Universität Kiel, Kiel, 1992, pp. 115-136.) showed that if the values satisfy a “single crossing” condition, then the two-bidder English auction has an efficient equilibrium. In this paper, two extensions of the single crossing condition, the “average crossing” condition and the “cyclical crossing” condition, are introduced. The main result is that under either of these conditions the n-bidder English auction has an efficient equilibrium.     

Information structures in optimal auctions

A seller wishes to sell an object to one of multiple bidders. The valuations of the bidders are privately known. We consider the joint design problem in which the seller can decide the accuracy by which bidders learn their valuation and to whom to sell at what price. We establish that optimal information structures in an optimal auction exhibit a number of properties: (i) information structures can be represented by monotone partitions, (ii) the cardinality of each partition is finite, (iii) the partitions are asymmetric across agents. We show that an optimal information structure exists.     

Existence of equilibrium in large double auctions

We show the existence of a pure strategy, symmetric, increasing equilibrium in double auction markets with correlated, conditionally independent private values and many participants. The equilibrium we find is arbitrarily close to fully revealing as the market size grows. Our results provide strategic foundations for price-taking behavior in large markets.     

Endogenous bid rotation in repeated auctions

I study collusion between two bidders in a general symmetric IPV repeated auction, without communication, side transfers, or public randomization. I construct a collusive scheme, endogenous bid rotation, that generates a payoff larger than the bid rotation payoff.     

Private information revelation in common-value auctions

When a seller has information that could help bidders to estimate asset value, a dictum of auction theory has been that all such information should be publicly announced to bidders. The possibility of privately revealing this information to one or more bidders is introduced. Seller in some circumstances may attain higher expected revenue through revealing his information privately. Examples show that the role of private revelation is more complex than simply generating bidder asymmetry.     

Maxmin mechanism in a simple common value auction

This paper models the data generating process of common value auctions in a parameter-estimation way, known as the classical approach in statistical inference. Viewing the true value of the object as a parameter that nobody ever knows, we let our value function be the average of the individual estimations (signals) of all agents, a robust estimation of the parameter. Under this simple value function, we select almost uniquely the mechanism that gives the seller the largest portion of the true value in the worst situation among all the direct mechanisms that are feasible, ex-post implementable and individual rational. Our Maxmin mechanism, which randomly assigns the object to one agent, provides the seller (n−1)/n$(n-1)/n$ of the true value when expected revenue is concerned, where n$n$ is the number of the agents.     

On optimal head starts in all-pay auctions

We consider a two-player all-pay auction with symmetric independent private values that are uniformly distributed. The designer chooses the size of a head start that is given to one of the players. The designer’s objective is to maximize a convex combination of the expected highest effort and the expected aggregate effort. Unless the weight on the highest effort is one, small head starts are always worse than no head start. Moreover, the optimal head start is strictly positive if and only if the weight on the highest effort is large enough.     

Revenue ranking of first-price auctions with resale

When the price setter in post-auction resale is chosen according to exogenous probabilities, Hafalir and Krishna (2008) [2] showed that the first-price auction brings more expected revenues than the second-price auction with truth-bidding bidders. We complete their revenue ranking by proving that the first-price auction produces higher expected revenues the higher the probability the auction winner sets the resale price.     

Optimal search auctions

We study the design of profit maximizing single unit auctions under the assumption that the seller needs to incur costs to contact prospective bidders and inform them about the auction. With independent bidders’ types and possibly interdependent valuations, the seller's problem can be reduced to a search problem in which the surplus is measured in terms of virtual utilities minus search costs. Compared to the socially efficient mechanism, the optimal mechanism features fewer participants, longer search conditional on the same set of participants, and inefficient sequence of entry.     

Mixed bundling auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.     

Equilibrium reserve prices in sequential ascending auctions

We study a model where bidders have perfectly correlated valuations for two goods sold sequentially in two ascending-price auctions. The seller sets a reserve price before the beginning of each auction. Despite the lack of commitment by the seller, we characterize an equilibrium and study its properties. Strategic non-disclosure of information takes the form of non-participation in the early auction by low-valuation bidders, while high-valuation bidders bid up to their true valuations. Some buyers who would profitably buy at the reserve price refrain from participating in order to decrease the second-auction reserve price.     

Multi-unit auctions with uniform prices

Summary. Auctions in which individuals can purchase more than one unit of the good being sold differ in striking ways from multi-unit auctions in which individuals may purchase only one unit. The uniform price auction in particular frequently yields Nash equilibria in which bidders underbid for their second unit and therefore pay very low prices for the good. This paper characterizes equilibria for the uniform price auction. Received: July 31, 1995; revised version: May 28, 1997     

Asymmetric information about rivals'' types in standard auctions

We study auctions in which bidders may know the types of some rival bidders but not others. This asymmetry in bidders' knowledge about rivals' types has different effects on the two standard auction formats. In a second-price auction, it is weakly dominant to bid one's valuation, so the knowledge of rivals' types has no effect, and the good is allocated efficiently. In a first-price auction, bidders refine their bidding strategies based on their knowledge of rivals' types, which yields an inefficient allocation. We show that the inefficient allocation in the first-price auction translates into a poor revenue performance. Given a standard regularity condition, the seller earns higher expected revenue from the second-price auction than from the first-price auction, whereas the bidders are better off from the latter.     

The value of an informed bidder in common value auctions

I compare two information structures in a common value first-price auction with two bidders: In one, each of the two bidders knows only his own signal about the value of the object, and in the other, one of the bidders learns his opponent's signal as well. Gaining the additional information in the second information structure makes the informed bidder worse off if the value is submodular in the bidders' signals and better off if it is supermodular. If the value is supermodular, then the seller's revenue tends to be lower with the informed bidder than without.     

Revenue equivalence in asymmetric auctions

The Revenue Equivalence Theorem is generalized to the case of asymmetric auctions in which each player's valuation is drawn independently from a common support according to his/her distribution function.